Posted by: Steven Sam | July 23, 2010

BGG resolutions and determinantal formulas

I want to discuss some character formulas that arise in Lie theory and symmetric functions, and how they are related to certain equivariant resolutions. A weird application of these resolutions is that certain partial alternating sums of symmetric functions are Schur positive. It’s unclear to me what that might imply. At first we’ll work in the general Kac–Moody setting, but one can imagine the example {\mathfrak{gl}_n} if that helps.

Weyl–Kac character formula.

Let {\mathfrak{g}} be a complex Kac–Moody algebra associated to a Cartan matrix {A = (a_{i,j})}, together with a Cartan subalgebra and Borel subalgebra {{\mathfrak h} \subset {\mathfrak b}}, and Weyl group {W}. Let {{\mathfrak b}^-} be the opposite Borel and set {{\mathfrak n} = [{\mathfrak b}, {\mathfrak b}]} and {{\mathfrak n}^- = [{\mathfrak b}^-, {\mathfrak b}^-]}. The simple coroots are {h_i \in {\mathfrak h}} and we define {\rho \in {\mathfrak h}^*} by the condition {\rho(h_i) = 1}. We’ll denote the set of dominant weights by {D} and the set of positive roots by {\Delta}. Given a positive root {\alpha \in \Delta}, let {m(\alpha)} be the dimension of the {\alpha}-root space of {\mathfrak{g}}. Also define the dotted action of {W} on the set of weights via {w \bullet \lambda = w(\lambda + \rho) - \rho}.

Given a dominant weight {\lambda \in D}, we have a unique irreducible representation {V_\lambda} of {\mathfrak{g}} with highest weight {\lambda}. Its character is the formal linear combination {\sum_\mu \dim V_\lambda(\mu) e^\mu} where {V_\lambda(\mu)} is the {\mu}-weight space. More compactly, this is given by the Weyl–Kac formula:

\displaystyle  {\rm char} V_\lambda = \sum_{i \ge 0} (-1)^i \sum_{w \in W, \ell(w) = i} \frac{e^{w \bullet \lambda}}{\prod_{\alpha \in \Delta} (1-e^{-\alpha})^{m(\alpha)}}.

I’ve written it in this way to suggest that the sign {(-1)^i} might be coming from the Euler characteristic of some complex. In fact, it does, which is what we will discuss next.

Given any integral weight {\mu}, we can form a highest weight module {M_\mu} as {U {\mathfrak{g}} / I_\mu}, where {U\mathfrak{g}} is the universal enveloping algebra of {\mathfrak{g}}, and {I_\mu} is the left ideal generated by {U{\mathfrak n}} and {x - \mu(x)} for all {x \in {\mathfrak h}}. This is known as the Verma module, and {M_\mu} is a free module over {U{\mathfrak n}^-}, so that its character is

\displaystyle  {\rm char} M_\mu = \frac{e^\mu}{\prod_{\alpha \in \Delta} (1-e^{-\alpha})^{m(\alpha)}}.

Looks familiar!

BGG resolutions.

The Bernstein–Gelfand–Gelfand (BGG) resolution for {V_\lambda} is the {\mathfrak{g}}-equivariant resolution

\displaystyle  \cdots \rightarrow C_i \rightarrow C_{i-1} \rightarrow \cdots \rightarrow C_1 \rightarrow C_0 \rightarrow V_\lambda \rightarrow 0,

where {C_i = \bigoplus_{w \in W, \ell(w) = i} M_{w \bullet \lambda}}. Note also that by looking at the {\mu}-weight space of this complex, we also get Kostant’s multiplicity formula (which makes sense when {\dim \mathfrak{g} < \infty})

\displaystyle  \dim V_\lambda(\mu) = \sum_{w \in W} (-1)^{\ell(w)} p(w \bullet \lambda - \mu),

where {p(\alpha)} is the number of ways to write {\alpha} as a sum of positive roots.

The construction for this resolution requires a few facts. First, given a dominant weight {\lambda \in D}, we have that {\dim {\rm Hom}_\mathfrak{g}(M_{w \bullet \lambda}, M_{w' \bullet \lambda}) \le 1}. Furthermore, the dimension is 1 if and only if {w \ge w'} in the Bruhat order, and all nonzero maps are injective. Hence we may fix an inclusion {M_{w \bullet \lambda} \subseteq M_\lambda = C_0} for all {w \in W}.

So for the map {C_i \rightarrow C_{i-1}}, we only need to pick a constant {s(w,w')} for the maps {M_{w \bullet \lambda} \rightarrow M_{w' \bullet \lambda}} when {w \ge w'}. There is no unique way to do this, but one such way comes from the following two facts. First, given {w_1, w_2} such that {\ell(w_2) = \ell(w_1) + 2}, either {w_1 \not\leq w_2}, or the interval {[w_1, w_2]} consists of 4 elements. Second, there exists a way to assign the numbers {\pm 1} to the covering relations in the Bruhat order of {W} such that in the interval {[w_1, w_2]} above, the product over the two maximal chains have opposite signs. Then we set {s(w,w') = \pm 1} according to how these signs are prescribed.

From these two facts, it is then clear that the maps just defined give us a complex. The exactness is more delicate, and I’ll refer the reader to Chapter IX of Kumar’s book Kac–Moody Groups, their Flag Varieties and Representation Theory for more details.

Zelevinsky’s functor.

For the second part of this post, I want to discuss a functor that Zelevinsky introduced in “Resolvents, dual pairs, and character formulas” that transforms the BGG resolution into other resolutions that realize the Jacobi–Trudi identity and its variants. A special case of this functor was considered by Akin in “On complexes relating the Jacobi-Trudi identity with the Bernstein-Gel’fand-Gel’fand resolution” for the Jacobi–Trudi identity. In fact, his construction for this case gives something more general, but I won’t discuss it. I won’t be needing W to denote the Weyl group anymore, so I’ll use it to name vector spaces (sorry!)

The functor {\Phi_{W,\nu}} goes from representations of {\mathfrak{g}} to vector spaces, and has two parameters. The first is a representation {W} of {\mathfrak{g}}, and the second is a weight {\nu}. It is defined as

\displaystyle  \Phi_{W, \nu}(V) = [(W \otimes V) / {{\mathfrak n}^-}(W \otimes V)](\mu).

The main properties are that the resulting complex obtained from applying {\Phi_{W,\nu}} to a BGG resolution is exact, and that

\displaystyle  \Phi_{W, \nu}(M_\mu) = W(\nu - \mu), \quad \Phi_{W, \nu}(V_\mu) = W(\nu - \mu, \mu),


\displaystyle  W(\nu - \mu, \mu) = \{ v \in W(\nu - \mu) \mid E_\alpha^{(\mu, \alpha^\vee) + 1} v = 0 \text{ for all simple roots } \alpha\}

and {E_\alpha} denotes a nonzero vector in the {\alpha}-root space of {\mathfrak{g}} and {(,)} is the Killing form. An important property of this subspace is that its dimension is the multiplicity of {V_\nu} in the tensor product {W \otimes V_\mu}.

Note that given the nature of the function {\Phi_{W, \nu}}, if {W} carries commuting actions of {\mathfrak{g}} and some other algebra {A}, then the functor {\Phi_{W,\nu}} goes to representations of {A}. Let’s give some examples.

Jacobi–Trudi identity.

Let {A} and {B} be two vector spaces. Let {W = {\rm Sym}^d(A \otimes B)}. The Cauchy identity gives the {\mathfrak{gl}(A) \times \mathfrak{gl}(B)}-equivariant decomposition

\displaystyle  W = \bigoplus_{\lambda \vdash d} {\bf S}_\lambda(A) \otimes {\bf S}_\lambda(B)

where {{\bf S}_\lambda(A)} is the irreducible representation of {\mathfrak{gl}(A)} of highest weight {\lambda} and similarly for {{\bf S}_\lambda(B)}. We take {\mathfrak{g} = \mathfrak{gl}(A)} and apply the functor {\Phi_{W, \lambda}} to the BGG resolution of {V_\mu}. We need to calculate {W(\lambda - w \bullet \mu)} and {W(\lambda - \mu, \mu)} to figure out what the resulting resolution is.

For the first one, let’s calculate {W(\chi)} more generally. Since we’re only interested in the action of {{\mathfrak h} \subset \mathfrak{gl}(A)}, we can write {A = {\bf C}e_1 \oplus \cdots {\bf C}e_n} where {\dim A = n} and {\{e_1, \dots, e_n\}} is the basis that diagonalizes {{\mathfrak h}}. For an {n}-tuple of nonnegative integers {{\bf d} = (d_1, \dots, d_n)}, {{\bf C}({\bf d})} is the 1-dimensional representation of {{\mathfrak h}} with character {{\bf d}}, and {{\rm Sym}^{\bf d}(B) = \bigotimes_{i=1}^n {\rm Sym}^{d_i}(B)}. Then we have

\displaystyle  {\rm Sym}^d(({\bf C}e_1 \oplus \cdots \oplus {\bf C}e_n) \otimes B) = \bigoplus_{\substack{{\bf d} = (d_1, \dots, d_n),\\ d_1 + \cdots + d_n = d,\\ d_i \ge 0}} {\bf C}({\bf d}) \otimes {\rm Sym}^{\bf d}(B),

so {W(\chi) = {\rm Sym}^\chi(B)}. Finally, what is {W(\lambda - \mu, \mu)}? Well, from what we mentioned above, {\dim {\bf S}_\nu(A)(\lambda - \mu, \mu)} is the multiplicity of {{\bf S}_\lambda} in {{\bf S}_\nu \otimes {\bf S}_\mu}, which is the Littlewood–Richardson coefficient {c_{\mu, \nu}^{\lambda}}. Hence

\displaystyle  W(\lambda - \mu, \mu) = \bigoplus_\nu {\bf S}_\nu(B)^{\oplus c^\lambda_{\mu, \nu}} = {\bf S}_{\lambda / \mu}(B),

where {{\bf S}_{\lambda / \mu}(B)} is a skew Schur functor, and the last equality follows from standard properties of skew Schur polynomials.

So the resulting resolution we get from applying {\Phi_{W, \lambda}} to the BGG resolution of {V_\mu} looks like {0 \rightarrow C_\bullet \rightarrow {\bf S}_{\lambda / \mu}(B) \rightarrow 0} where

\displaystyle  C_i = \bigoplus_{w \in S_n, \ell(w) = i} {\rm Sym}^{\lambda - w \bullet \mu}(B)

({n = \dim A}). In this case, {\rho = (n-1, n-2, \dots, 1, 0)}. Taking the {\mathfrak {gl}(B)}-equivariant Euler characteristic, we get

\displaystyle  s_{\lambda / \mu}(x) = \sum_{w \in S_n} (-1)^{\ell(w)} h_{\lambda - w(\mu + \rho) - \rho}(x) = \det(h_{\lambda_i - \mu_j - i + j}(x))_{i,j=1}^n,

which is the Jacobi–Trudi identity. If we replace the symmetric powers above with exterior powers (and use the dual Cauchy formula), we’ll get the dual Jacobi–Trudi identity.

Determinantal expression for products of Schur functions.

A slight variation gives a not as well known determinantal formula: we take {W} as above, but apply {\Phi_{W, \lambda}} to the BGG resolution of {V_\mu^* = V_{\mu^*}} where {\mu^* = (-\mu_n, -\mu_{n-1}, \dots, -\mu_1)}. Then {\dim {\bf S}_\nu(A)(\lambda - \mu^*, \mu^*)} is the multiplicity of {{\bf S}_\lambda(A)} in {{\bf S}_\nu(A) \otimes {\bf S}_\mu(A)^*}, which is equal to

\displaystyle  \dim({\bf S}_\lambda(A)^* \otimes {\bf S}_\nu(A)^* \otimes {\bf S}_\mu(A)^*)^{\mathfrak{gl}(A)}.

The space of invariants is unchanged by taking a dual, so we see that this dimension is the multiplicity of {{\bf S}_\nu(A)} in {{\bf S}_\lambda(A) \otimes {\bf S}_\mu(A)}, or the Littlewood–Richardson coefficient {c^\nu_{\lambda, \mu}}, so

\displaystyle  W(\lambda - \mu^*, \mu^*) = \bigoplus_\nu {\bf S}_\nu(B)^{\oplus c^{\nu}_{\lambda, \mu}} = {\bf S}_\lambda(B) \otimes {\bf S}_\mu(B),

and hence we’re now resolving a tensor product. This time the {\mathfrak{gl}(B)}-equivariant Euler characteristic is

\displaystyle  s_\lambda(x) s_\mu(x) = \sum_{w \in S_n} (-1)^{\ell(w)} h_{\lambda - w(\mu^* + \rho) - \rho}(x) = \det(h_{\lambda_i + \mu_{n+1-j} - i + j}(x))_{i,j=1}^n.

Note that the product {s_\lambda(x)s_\mu(x)} is equal to a skew Schur function by making a shape where the diagrams of {\lambda} and {\mu} don’t have any boxes in the same row or column, but this will in general have {2n} rows, so this determinantal expression is more efficient.

Symmetric groups.

For an extra formula, take {W = A^{\otimes k}} with the action of {\mathfrak{gl}(A)} and the action of {S_k} on the tensor factors. Then Schur–Weyl duality gives

\displaystyle  W = \bigoplus_\lambda {\bf S}_\lambda(A) \otimes \chi_\lambda

where {\chi_\lambda} index irreducible representations of {S_k}. Apply {\Phi_{W, \lambda}} to the BGG resolution of {V_\mu} over {\mathfrak{gl}(A)} and see what you get.



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